In this paper we consider a nonlinear parabolic equation of the following type:
(P) ∂u/∂t - div(|∇p|p-2 ∇u) = h(x,u)
with Dirichlet boundary conditions and initial data in the case when 1 &lt; p &lt; 2.
We construct supersolutions of (P), and by use of them, we prove that for tn → +∞, the solution of (P) converges to some solution of the elliptic equation associated with (P).

We study the existence of solutions of the nonlinear parabolic problem
∂u/∂t-div[|Du-Θ(u)|p-2(Du-Θ(u))]+α(u)=f in ]0,T[ × Ω,
(|Du-Θ(u)|p-2(Du-Θ(u)))·η+γ(u)=g on ]0,T[ × ∂Ω,
u(0,·) = u₀ in Ω,
with initial data in L¹. We use a time discretization of the continuous problem by the Euler forward scheme.